Question

Implicit differentiation involving trig and finding equation of tangent line. Please help

Answer 1

\[ysin(2x) = xcos(2y)\] . find equation of tangent line at the point \[\frac{ \pi }{ 2 }, \frac{ \pi }{ 4 }\]

Answer 2

so, if you find dy/dx and then put x = pi/2, y= pi/4
you will get slope of tangent at that point.
now you have slope and a point on a line, it would be easy to find its equation.

Answer 3

i solved for dy/dx. unsure of how to proceed from there to get the gradient of my equation of the tangent line.

Answer 4

put x = pi/2, y= pi/4

Answer 5

yeah i know that, how would i solve for pi/2 and pi/4 for example cos(pi/2) ?

Answer 6

cos (pi/2) = 0

Answer 7

here is my dy/dx:

\[\frac{ dy }{ dx } = \frac{ \cos(2y) - 2y.\cos(2x) }{ \sin(2x) + 2x.\sin(2y) }\]

Answer 8

and its absolutely correct :)

Answer 9

so i would plug in for x and y respectivley with the point given?

Answer 10

yes.

Answer 11

what if it doesnt factor out nicely would I still use it as my m for y=mx+c

Answer 12

going to solve for x,y now, gimme 2min please :)

Answer 13

sure, take your time :)

Answer 14

\(\huge m = \frac{ \cos(\pi/2) - 2(\pi/4).\cos(\pi) }{ \sin(\pi) + 2(\pi/2).\sin(\pi/2) }\)

Answer 15

slight mistake in your cos(pi/2).

Should be cos(2.pi/2) which can be simplified to cos(pi).

correct?

Answer 16

y=pi/4

Answer 17

plugged in for points pi/2 and pi/4. Then simplified further.

Then used trig to find the arc of the sin/cos to get a number.

\[\frac{ \cos(\pi)-\frac{ \pi }{ 2 }\cos(\pi) }{ \sin(\pi) + \pi.\sin(\frac{ \pi }{ 2 }) }\]

Answer 18

y = pi/4
cos 2y = cos 2*pi/4 = cos pi/2

Answer 19

ahh yes you are correct.

Answer 20

so, m=.. ?

Answer 21

\[\frac{ \cos(\ \frac{ \pi }{ 2 })-\frac{ \pi }{ 2 }\cos(\pi) }{ \sin(\pi) + \pi.\sin(\frac{ \pi }{ 2 }) }\]

Answer 22

simplify that, can u ?

Answer 23

solved further using trig:

\[\frac{ -1-\frac{ \pi }{ 2 } (-1)}{ 0+\pi(1)}\]

Answer 24

please double check that for me

Answer 25

cos pi/2 = 0

Answer 26

everything else is correct

Answer 27

ah yes, forgot to edit my initial calculation

Answer 28

m = 1/2

Answer 29

yes :)

Answer 30

ahhh.

now I simply use y = mx + c.

y = 0.5(x)+c

plugin points pi/2, pi/4 for x and y respectivley:

pi/4 = 0.5(pi/2) + c
c = 0.5

Answer 31

\[\therefore y=\frac{ 1 }{ 2 }x\]

Answer 32

you could also use
y- y1 = m (x-x1)
y-pi/4 = 0.5 (x- pi/2)
y- pi/4 = 0.5 x - pi/4
y= x/2
yes, its correct :)

Answer 33

thanks so much once again!

Answer 34

welcome once again ^_^